Theorem wwlktovf1 13548

Description: Lemma 2 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 27-Jul-2018.)

Hypotheses

Ref	Expression
wrd2f1tovbij.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wrd2f1tovbij.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wrd2f1tovbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))

Assertion

Ref	Expression
wwlktovf1	⊢ 𝐹:𝐷–1-1→𝑅

Distinct variable groups:   𝑡,𝐷   𝑃,𝑛,𝑡,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑋,𝑤

Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐹(𝑤,𝑡,𝑛)   𝑋(𝑡)

Proof of Theorem wwlktovf1

Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrd2f1tovbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2		wrd2f1tovbij.r	. . 3 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3		wrd2f1tovbij.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))
4	1, 2, 3	wwlktovf 13547	. 2 ⊢ 𝐹:𝐷⟶𝑅
5		fveq1 6102	. . . . . 6 ⊢ (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6		fvex 6113	. . . . . 6 ⊢ (𝑥‘1) ∈ V
7	5, 3, 6	fvmpt 6191	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥‘1))
8		fveq1 6102	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9		fvex 6113	. . . . . 6 ⊢ (𝑦‘1) ∈ V
10	8, 3, 9	fvmpt 6191	. . . . 5 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦‘1))
11	7, 10	eqeqan12d 2626	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12		fveq2 6103	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
13	12	eqeq1d 2612	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((#‘𝑤) = 2 ↔ (#‘𝑥) = 2))
14		fveq1 6102	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
15	14	eqeq1d 2612	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
16		fveq1 6102	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
17	14, 16	preq12d 4220	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
18	17	eleq1d 2672	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
19	13, 15, 18	3anbi123d 1391	. . . . . 6 ⊢ (𝑤 = 𝑥 → (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20	19, 1	elrab2 3333	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
21		fveq2 6103	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (#‘𝑤) = (#‘𝑦))
22	21	eqeq1d 2612	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((#‘𝑤) = 2 ↔ (#‘𝑦) = 2))
23		fveq1 6102	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
24	23	eqeq1d 2612	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
25		fveq1 6102	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
26	23, 25	preq12d 4220	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
27	26	eleq1d 2672	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
28	22, 24, 27	3anbi123d 1391	. . . . . 6 ⊢ (𝑤 = 𝑦 → (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
29	28, 1	elrab2 3333	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
30		simp1 1054	. . . . . . . . . 10 ⊢ (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (#‘𝑥) = 2)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (#‘𝑥) = 2)
32		simp1 1054	. . . . . . . . . . 11 ⊢ (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (#‘𝑦) = 2)
33	32	eqcomd 2616	. . . . . . . . . 10 ⊢ (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 2 = (#‘𝑦))
34	33	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (#‘𝑦))
35	31, 34	sylan9eq 2664	. . . . . . . 8 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (#‘𝑥) = (#‘𝑦))
36	35	adantr 480	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (#‘𝑥) = (#‘𝑦))
37		simp2 1055	. . . . . . . . . . 11 ⊢ (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (𝑥‘0) = 𝑃)
38	37	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
39		simp2 1055	. . . . . . . . . . . 12 ⊢ (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → (𝑦‘0) = 𝑃)
40	39	eqcomd 2616	. . . . . . . . . . 11 ⊢ (((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋) → 𝑃 = (𝑦‘0))
41	40	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
42	38, 41	sylan9eq 2664	. . . . . . . . 9 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥‘0) = (𝑦‘0))
43	42	adantr 480	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘0) = (𝑦‘0))
44		simpr 476	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘1) = (𝑦‘1))
45		oveq2 6557	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) = 2 → (0..^(#‘𝑥)) = (0..^2))
46		fzo0to2pr 12420	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
47	45, 46	syl6eq 2660	. . . . . . . . . . . 12 ⊢ ((#‘𝑥) = 2 → (0..^(#‘𝑥)) = {0, 1})
48	47	raleqdv 3121	. . . . . . . . . . 11 ⊢ ((#‘𝑥) = 2 → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖)))
49		c0ex 9913	. . . . . . . . . . . 12 ⊢ 0 ∈ V
50		1ex 9914	. . . . . . . . . . . 12 ⊢ 1 ∈ V
51		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
52		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
53	51, 52	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
54		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑥‘𝑖) = (𝑥‘1))
55		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑦‘𝑖) = (𝑦‘1))
56	54, 55	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
57	49, 50, 53, 56	ralpr 4185	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
58	48, 57	syl6bb 275	. . . . . . . . . 10 ⊢ ((#‘𝑥) = 2 → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
59	58	3ad2ant1 1075	. . . . . . . . 9 ⊢ (((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
60	59	ad3antlr 763	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
61	43, 44, 60	mpbir2and 959	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))
62		simpl 472	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → 𝑥 ∈ Word 𝑉)
63		simpl 472	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑦 ∈ Word 𝑉)
64	62, 63	anim12i 588	. . . . . . . . 9 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉))
65	64	adantr 480	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉))
66		eqwrd 13201	. . . . . . . 8 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
67	65, 66	syl 17	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ∀𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
68	36, 61, 67	mpbir2and 959	. . . . . 6 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → 𝑥 = 𝑦)
69	68	ex 449	. . . . 5 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((#‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
70	20, 29, 69	syl2anb 495	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
71	11, 70	sylbid 229	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
72	71	rgen2a 2960	. 2 ⊢ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
73		dff13 6416	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
74	4, 72, 73	mpbir2an 957	1 ⊢ 𝐹:𝐷–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {cpr 4127   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  ..^cfzo 12334  #chash 12979  Word cword 13146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154

This theorem is referenced by:  wwlktovf1o  13550